Non-Stationary Noise

Hi Everyone,


I am trying to derive a theoretical curve for noise reduction under
non-stationary condition and see how that compare with my data and
computer simulations.


Here is the gist of the problem. I have a measured signal which is
composed of a deterministic component (s) and some nonstationary noise
(n(t)):

x(t)= s + n(t)


To simplifify things, we can assume that the noise n(t) is stationary,
white-zero mean  with variance sig^2  over brief intervals (Blocks).
With these assumptions in place we know that if we apply standard
averaging to x(t) the noise reduction in the first block should be
sig^2/ N (where N is the number of trials you average over with):

var(mx(t)) = var(s) + sig^2/N     (assuming n(t) is uncorrelated with
s)

Where mx(t) is the mean over N sample waveforms.
Now the problem I am getting stuck is continuing this reasoning with
the next Blocks. Because I am using a running meaning:

mx(t) =( (N-1)*mx(t-1) + xn(t) ) /N            n=1...N


I want to me to come with up with a theoretical curve for the variance
of mx(t) across blocks of stationary noise but with different
variances (sig^2).


Computer simulations give me a feel that the variance of mx(t) behave
in simlar manner to a excitation/decay of a LTC system in between
blocks....

I hope this is clear...

Any help and links would be very useful !!

Thank you

On Tue, 30 Oct 2007 09:29:11 -0700, Ikaro wrote:

> Hi Everyone,
> 
> 
> I am trying to derive a theoretical curve for noise reduction under
> non-stationary condition and see how that compare with my data and
> computer simulations.
> 
> 
> Here is the gist of the problem. I have a measured signal which is
> composed of a deterministic component (s) and some nonstationary noise
> (n(t)):
> 
> x(t)= s + n(t)
> 
> 
> To simplifify things, we can assume that the noise n(t) is stationary,
> white-zero mean  with variance sig^2  over brief intervals (Blocks).
> With these assumptions in place we know that if we apply standard
> averaging to x(t) the noise reduction in the first block should be
> sig^2/ N (where N is the number of trials you average over with):
> 
> var(mx(t)) = var(s) + sig^2/N     (assuming n(t) is uncorrelated with
> s)
> 
> Where mx(t) is the mean over N sample waveforms.
> Now the problem I am getting stuck is continuing this reasoning with
> the next Blocks. Because I am using a running meaning:
> 
> mx(t) =( (N-1)*mx(t-1) + xn(t) ) /N            n=1...N
> 
> 
> I want to me to come with up with a theoretical curve for the variance
> of mx(t) across blocks of stationary noise but with different
> variances (sig^2).
> 
> 
> Computer simulations give me a feel that the variance of mx(t) behave
> in simlar manner to a excitation/decay of a LTC system in between
> blocks....
> 
> I hope this is clear...
> 
> Any help and links would be very useful !!
> 
> Thank you

Kalman filters do a good job with time-varying systems.  If your noise is
zero-mean Gaussian with a known relationship between time and power then
you can frame your system as having a constant input noise power but with
a varying noise coupling matrix.  If you do this the Kalman matricies
should just drop out.

Unfortunately the only reference I have that I know for sure will cover
this is Van Trees, "Detection, Estimation and Modulation Theory, Vol.
I". It's not a terribly accessible book.  In fact, without a good prof to
help you out you'll probably never quite get it -- but it'll give you
foundations that go down to bedrock.  You may also want to try Simon,
"Optimal State Detection".  It's a _lot_ more accessible that the Van
Trees book, but I have only skimmed it so I can't guarantee that it'll
teach you this part of Kalman filtering.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Hi Tim,


Thanks I will check the Van Trees book that you mentioned.
I am not sure, but I have a feeling that the Kalman approach might be
an overkill...

I just looking for a closed-form equation for the background noise
power under normal averaging method when the noise power is
nonstationary (stationary only over short periods)...

Thank Again for the last post.

(restored context here)

On Tue, 30 Oct 2007 09:29:11 -0700, Ikaro wrote:

> Hi Everyone,
> 
> 
> I am trying to derive a theoretical curve for noise reduction under
> non-stationary condition and see how that compare with my data and
> computer simulations.
> 
> 
> Here is the gist of the problem. I have a measured signal which is
> composed of a deterministic component (s) and some nonstationary noise
> (n(t)):
> 
> x(t)= s + n(t)
> 
> 
> To simplifify things, we can assume that the noise n(t) is stationary,
> white-zero mean  with variance sig^2  over brief intervals (Blocks).
> With these assumptions in place we know that if we apply standard
> averaging to x(t) the noise reduction in the first block should be
> sig^2/ N (where N is the number of trials you average over with):
> 
> var(mx(t)) = var(s) + sig^2/N     (assuming n(t) is uncorrelated with
> s)
> 
> Where mx(t) is the mean over N sample waveforms.
> Now the problem I am getting stuck is continuing this reasoning with
> the next Blocks. Because I am using a running meaning:
> 
> mx(t) =( (N-1)*mx(t-1) + xn(t) ) /N            n=1...N
> 
> 
> I want to me to come with up with a theoretical curve for the variance
> of mx(t) across blocks of stationary noise but with different
> variances (sig^2).
> 
> 
> Computer simulations give me a feel that the variance of mx(t) behave
> in simlar manner to a excitation/decay of a LTC system in between
> blocks....
> 
> I hope this is clear...
> 
> Any help and links would be very useful !!
> 
> Thank you

(end restored context)

On Tue, 30 Oct 2007 11:32:15 -0700, Ikaro wrote:

> Hi Tim,
> 
> 
> Thanks I will check the Van Trees book that you mentioned.
> I am not sure, but I have a feeling that the Kalman approach might be
> an overkill...
> 
> I just looking for a closed-form equation for the background noise
> power under normal averaging method when the noise power is
> nonstationary (stationary only over short periods)...
> 
> Thank Again for the last post.

Are you analyzing the effect of a given filter, or are you trying to
find the best filter?  If it's "best" you're looking for, then you need to
go down the Kalman road (and please find an easier book than Van Trees --
it's what I learned out of, but I wouldn't recommend it at all unless
you're taking a class with a _good_ prof).

If you just want to know the effect of your filter then tot up the noise
variance of each block, find the contribution of each block to your
current output, and use the two to calculate the noise variance at your
current output.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Hi Tim,


I am not using any filter at all. I just wanted to characterize how
the SNR behaves under normal averaging but with non-stationary noise
sources. Offcourse, under stationary condition the SNR increases by
n^0.5, but things get more complicated for non-stationary noise...


Fortunately, I was able to derive a closed form equation
characterizing the non-stationary case (assuming blocks of
stationarity and a diagonal form of the covariance matrix for the
noise).


Thanks for the help !


-Ikaro

On Fri, 02 Nov 2007 12:25:51 -0700, Ikaro wrote:

> Hi Tim,
> 
> 
> I am not using any filter at all. I just wanted to characterize how
> the SNR behaves under normal averaging but with non-stationary noise
> sources. Offcourse, under stationary condition the SNR increases by
> n^0.5, but things get more complicated for non-stationary noise...
> 
> 
> Fortunately, I was able to derive a closed form equation
> characterizing the non-stationary case (assuming blocks of
> stationarity and a diagonal form of the covariance matrix for the
> noise).
> 
> 
> Thanks for the help !
> 
> 
> -Ikaro

Averaging _is_ filtering!

At any rate, I'm glad that I helped.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html